# Tagged Questions

Questions about partial differential equations of parabolic type. Often used in combination with the top-level tag ap.analysis-of-pdes.

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### Properties of the solution of the heat equation

Note 1: the following question has been post on Math Stackexchange here but receive no respond. So I post it here to get more attention. Note 2: This is my research problem, but the original problem ...
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### Reference for short time existence of paraobolic PDE on bundles

I am looking for a reference treating parabolic equations on vector bundles. In particular, I look for precise conditions which guarantee short time existence. I found it quoted at different places in ...
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### an inverse problem related to gaussian integral

Suppose we have a function $\rho(x,t)$ defined over $[-1,1]\times[0,1]$. Define the integral $f_T(x)=\int_{[0,1]} (\rho(.,t)*K_{T-t})(x) dt$ for $x\in R$ and $T>1$, where $*$ is the convolution, ...
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### Reference for existence results for 2D forced viscous Burgers equation

I am looking for results concering the following parabolic PDE $$u\cdot\nabla u + \Delta u = F(x),$$ where $$u\colon\Omega\to\mathbb{R}^2,$$ and $\Omega\subset\mathbb{R}^2$ is a 2D domain (bounded or ...
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### Regularity of solution to Fokker Planck equation

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE \begin{align} \partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\ \rho(t =...
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### Weighted energy estimate for the heat equation of higher order

The question is originally related to Hardy's uncertainty principle, convexity and Schrodinger evolutions. In this work the authors deduce a convex property of Schrodinger equation by doing it first ...
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### regularity for viscosity solutions of second order parabolic equations

I would like to know whether viscosity solutions to $u_{t} - F( D^{2} (u) ) = 0$ are $C^{1, \alpha}$ analogous to the elliptic case as in the book by Caffarelli and Cabre . Here F is ...
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### $C^{1,2}$ regularity of (weak) solutions to the heat equation

Let $\Omega$ be a bounded Lipschitz domain (smoother if needed), and consider the heat equation $$u_t - \Delta u = 0$$ $$\frac{\partial u(t,x)}{\partial \nu(x)} = a(t,x) - b(t,x)u(t,x)$$ $$u(0) = u_0$$...
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### About the “method of lines”: when are such solutions good approximations for **all** future time?

This question is about approximate solutions to some classes of PDEs obtained using the "method of lines". For example, for an initial-value problem given by a PDE on a circle, one can choose $n$ ...
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### Cauchy Problem and stochastic representation for discontinuous initial data

Where can I read more about the Cauchy problem, i.e. solutions to $$\frac{\partial u}{\partial t}+Lu=0 \text{ and } u(0,x)=f(x)$$ for some elliptic differential operator $L$ where $f$ is not ...
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### References Request : Existence and Uniqueness for PDE which is “ALMOST (?)” Parabolic

I am doing a research using PDE which is a little different from the standard Parabolic type. The following is my case: Lu = - \sum_{i, j = 1}^{N}a^{i,j}(x, t)u_{x_{i}x_{j}} + \sum_{...
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I'm interested in the default Stokes-system, e.g. $\frac{\partial}{\partial t} u - \Delta u + \nabla p = f \; \text{in} \; \Omega$ $\nabla \cdot u = 0 \; \text{in} \; \Omega$ $u = 0 \; \text{on} ... 0answers 60 views ### Find a estimate for quasilinear parabolic equation I am studying quasilinear parabolic operator:$Pu=-u_t+u_{xx}+a(x;t;u,u_x)$where$(x,t)\in \Omega=(0,\pi)\times(0,T)$and$za(x,t,z,0)\le kz^2+b$Suppose that$u$satisfies:$P(u)=0$in$\Omega.$... 0answers 54 views ### Definiteness and infinite divisibility of kernels including heat semigroup Let$P_{t}$be the usual heat semigroup. Can one show (preferably) or disprove that for arbitrary$k \in \mathbb{R}_{>0}$and$n \in \mathbb{N}$we have \sum_{i,j=1}^{n}a_{i}a_{j}\... 0answers 88 views ### An Estimation on the Heat Kernel I am reading Jost's Partial Differential Equations and meet an estimation( only stated in the book ) which I cannot verify by myself. In p.111, the book says that iteratively, we get $$|S_{n}(x_{0},y,... 0answers 48 views ### Weak solutions of linear parabolic PDEs and corresponding SDEs It is well known that for an Stochastic differential equation (on the real line) of the form: dX_t = \mu(X_t)dt + \sigma(X_t)dW where W is the standard Wiener process, the transition probability ... 0answers 126 views ### Regularity of the heat equation: Neumann boundary conditions I am looking for references to regularity estimates for the solution of a heat equation with homogeneous Neumann boundary conditions on [0,T]\times D for some smooth domain D\subseteq \mathbb{R}^3 ... 0answers 79 views ### A heat equation approach to the perturbation of vector field with center Edit: According to the comment of Willie Wong I realize that the previous version was trivial. I thank him for his comment. Now I revise it. We consider the heat equation$$U_{t}=\Delta U\\U(x,y,0)=... 0answers 48 views ### Is the heat kernel satisfies the heat equation in viscosity sense? Let us see the heat kernel $$k(x,t)= \begin{cases} (4\pi t)^{-\frac{n}{2}}\exp\{-\frac{1}{4}t^{-1}|x|^{2}\},t>0\\ 0,t\leq0. \end{cases}$$ It is easy to see that$k\in C^{...
We consider the stationary navier stokes equation with mixed boundary conditions  \begin{align} -\nu\Delta u +(u\cdot\nabla) u + \nabla p &= 0\ \textrm{in}\ \Omega \\ div\ u&=0\ \textrm{in}\ ...